Theorem natoppfb 50009

Description: A natural transformation is natural between opposite functors, and vice versa. (Contributed by Zhi Wang, 18-Nov-2025.)

Hypotheses

Ref	Expression
natoppf.o	⊢ 𝑂 = (oppCat‘𝐶)
natoppf.p	⊢ 𝑃 = (oppCat‘𝐷)
natoppf.n	⊢ 𝑁 = (𝐶 Nat 𝐷)
natoppf.m	⊢ 𝑀 = (𝑂 Nat 𝑃)
natoppfb.k	⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
natoppfb.l	⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
natoppfb.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
natoppfb.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
natoppfb	⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Proof of Theorem natoppfb

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		natoppf.o	. . . 4 ⊢ 𝑂 = (oppCat‘𝐶)
2		natoppf.p	. . . 4 ⊢ 𝑃 = (oppCat‘𝐷)
3		natoppf.n	. . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		natoppf.m	. . . 4 ⊢ 𝑀 = (𝑂 Nat 𝑃)
5		natoppfb.k	. . . . 5 ⊢ (𝜑 → 𝐾 = ( oppFunc ‘𝐹))
6	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐾 = ( oppFunc ‘𝐹))
7		natoppfb.l	. . . . 5 ⊢ (𝜑 → 𝐿 = ( oppFunc ‘𝐺))
8	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝐿 = ( oppFunc ‘𝐺))
9		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐹𝑁𝐺))
10	1, 2, 3, 4, 6, 8, 9	natoppf2 50008	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹𝑁𝐺)) → 𝑥 ∈ (𝐿𝑀𝐾))
11		eqid 2763	. . . . 5 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
12		eqid 2763	. . . . 5 ⊢ (oppCat‘𝑃) = (oppCat‘𝑃)
13		eqid 2763	. . . . 5 ⊢ ((oppCat‘𝑂) Nat (oppCat‘𝑃)) = ((oppCat‘𝑂) Nat (oppCat‘𝑃))
14	7	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 = ( oppFunc ‘𝐺))
15	14	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐿) = ( oppFunc ‘( oppFunc ‘𝐺)))
16	4	natrcl 18005	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐿𝑀𝐾) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
17	16	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐿 ∈ (𝑂 Func 𝑃) ∧ 𝐾 ∈ (𝑂 Func 𝑃)))
18	17	simpld 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐿 ∈ (𝑂 Func 𝑃))
19	14, 18	eqeltrrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐺) ∈ (𝑂 Func 𝑃))
20		relfunc 17914	. . . . . . 7 ⊢ Rel (𝑂 Func 𝑃)
21		eqid 2763	. . . . . . 7 ⊢ ( oppFunc ‘𝐺) = ( oppFunc ‘𝐺)
22	19, 20, 21	2oppf 49910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐺)) = 𝐺)
23	15, 22	eqtr2d 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐺 = ( oppFunc ‘𝐿))
24	5	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 = ( oppFunc ‘𝐹))
25	24	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐾) = ( oppFunc ‘( oppFunc ‘𝐹)))
26	17	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐾 ∈ (𝑂 Func 𝑃))
27	24, 26	eqeltrrd 2864	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘𝐹) ∈ (𝑂 Func 𝑃))
28		eqid 2763	. . . . . . 7 ⊢ ( oppFunc ‘𝐹) = ( oppFunc ‘𝐹)
29	27, 20, 28	2oppf 49910	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → ( oppFunc ‘( oppFunc ‘𝐹)) = 𝐹)
30	25, 29	eqtr2d 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 = ( oppFunc ‘𝐾))
31		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐿𝑀𝐾))
32	11, 12, 4, 13, 23, 30, 31	natoppf2 50008	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
33	1	2oppchomf 17775	. . . . . . . 8 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
34	33	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
35	1	2oppccomf 17776	. . . . . . . 8 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))
36	35	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
37	2	2oppchomf 17775	. . . . . . . 8 ⊢ (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃))
38	37	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (Homf ‘𝐷) = (Homf ‘(oppCat‘𝑃)))
39	2	2oppccomf 17776	. . . . . . . 8 ⊢ (compf‘𝐷) = (compf‘(oppCat‘𝑃))
40	39	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (compf‘𝐷) = (compf‘(oppCat‘𝑃)))
41		natoppfb.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
42	41	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ 𝑉)
43		natoppfb.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
44	43	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ 𝑊)
45	1, 2, 42, 44, 27	funcoppc5 49923	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐹 ∈ (𝐶 Func 𝐷))
46	45	func1st2nd 49854	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹))
47	46	funcrcl2 49857	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐶 ∈ Cat)
48	1	oppccat 17773	. . . . . . . 8 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
49	11	oppccat 17773	. . . . . . . 8 ⊢ (𝑂 ∈ Cat → (oppCat‘𝑂) ∈ Cat)
50	47, 48, 49	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑂) ∈ Cat)
51	46	funcrcl3 49858	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝐷 ∈ Cat)
52	2	oppccat 17773	. . . . . . . 8 ⊢ (𝐷 ∈ Cat → 𝑃 ∈ Cat)
53	12	oppccat 17773	. . . . . . . 8 ⊢ (𝑃 ∈ Cat → (oppCat‘𝑃) ∈ Cat)
54	51, 52, 53	3syl 19	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (oppCat‘𝑃) ∈ Cat)
55	34, 36, 38, 40, 47, 50, 51, 54	natpropd 18031	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐶 Nat 𝐷) = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
56	3, 55	eqtrid 2810	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑁 = ((oppCat‘𝑂) Nat (oppCat‘𝑃)))
57	56	oveqd 7427	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → (𝐹𝑁𝐺) = (𝐹((oppCat‘𝑂) Nat (oppCat‘𝑃))𝐺))
58	32, 57	eleqtrrd 2866	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐿𝑀𝐾)) → 𝑥 ∈ (𝐹𝑁𝐺))
59	10, 58	impbida 812	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝐹𝑁𝐺) ↔ 𝑥 ∈ (𝐿𝑀𝐾)))
60	59	eqrdv 2761	1 ⊢ (𝜑 → (𝐹𝑁𝐺) = (𝐿𝑀𝐾))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  1^st c1st 7980  2^nd c2nd 7981  Catccat 17715  Hom_f chomf 17717  comp_fccomf 17718  oppCatcoppc 17762   Func cfunc 17906   Nat cnat 17996   oppFunc coppf 49900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-dec 12707  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-hom 17329  df-cco 17330  df-cat 17719  df-cid 17720  df-homf 17721  df-comf 17722  df-oppc 17763  df-func 17910  df-nat 17998  df-oppf 49901

This theorem is referenced by:  fucoppclem  50185  lmddu  50445